The function $f(x)$ satisfies
\[f(x + f(x)) = 4f(x)\]for all $x,$ and $f(1) = 4.$  Find $f(21).$
Explanation: Setting $x = 1,$ we get $f(1 + f(4)) = 4f(1),$ so
\[f(5) = 16.\]Setting $x = 5,$ we get $f(5 + f(5)) = 4f(5),$ so
\[f(21) = \boxed{64}.\]